Model theory recognises and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. To quote the first page of Chang & Keisler (1990):[1]

Informally, model theory can be divided into classical model theory, model theory applied to groups and fields, and geometric model theory. A missing subdivision is computable model theory, but this can arguably be viewed as an independent subfield of logic.

During the last several decades applied model theory has repeatedly merged with the more pure stability theory. The result of this synthesis is called geometric model theory in this article (which is taken to include o-minimality, for example, as well as classical geometric stability theory). An example of a theorem from geometric model theory is Hrushovski's proof of the Mordell–Lang conjecture for function fields. The ambition of geometric model theory is to provide a geography of mathematics by embarking on a detailed study of definable sets in various mathematical structures, aided by the substantial tools developed in the study of pure model theory.

Fundamental concepts in universal algebra are signatures σ and σ-algebras. Since these concepts are formally defined in the article on structures, the present article is an informal introduction which consists of examples of the way these terms are used.

The standard signature of rings is σring = {×,+,−,0,1}, where × and + are binary, − is unary, and 0 and 1 are nullary.

The standard signature of semirings is σsmr = {×,+,0,1}, where the arities are as above.

The standard signature of groups (with multiplicative notation) is σgrp = {×,−1,1}, where × is binary, −1 is unary and 1 is nullary.

This is a very efficient way to define most classes of algebraic structures, because there is also the concept of σ-homomorphism, which correctly specializes to the usual notions of homomorphism for groups, semigroups, magmas and rings. For this to work, the signature must be chosen well.

Terms such as the σring-term t(u,v,w) given by (u + (v × w)) + (−1) are used to define identities t = t', but also to construct free algebras. An equational class is a class of structures which, like the examples above and many others, is defined as the class of all σ-structures which satisfy a certain set of identities. Birkhoff's theorem states:

A class of σ-structures is an equational class if and only if it is not empty and closed under subalgebras, homomorphic images, and direct products.

An important non-trivial tool in universal algebra are ultraproductsΠi∈IAi/U{\displaystyle \Pi _{i\in I}A_{i}/U}, where I is an infinite set indexing a system of σ-structures Ai, and U is an ultrafilter on I.

Finite model theory is the area of model theory which has the closest ties to universal algebra. Like some parts of universal algebra, and in contrast with the other areas of model theory, it is mainly concerned with finite algebras, or more generally, with finite σ-structures for signatures σ which may contain relation symbols as in the following example:

The standard signature for graphs is σgrph={E}, where E is a binary relation symbol.

A σ-homomorphism is a map that commutes with the operations and preserves the relations in σ. This definition gives rise to the usual notion of graph homomorphism, which has the interesting property that a bijective homomorphism need not be invertible. Structures are also a part of universal algebra; after all, some algebraic structures such as ordered groups have a binary relation <. What distinguishes finite model theory from universal algebra is its use of more general logical sentences (as in the example above) in place of identities. (In a model-theoretic context an identity t=t' is written as a sentence ∀u1u2…un(t=t′){\displaystyle \forall u_{1}u_{2}\dots u_{n}(t=t')}.)

The logics employed in finite model theory are often substantially more expressive than first-order logic, the standard logic for model theory of infinite structures.

A first-order formula is built out of atomic formulas such as R(f(x,y),z) or y = x + 1 by means of the Boolean connectives¬,∧,∨,→{\displaystyle \neg ,\land ,\lor ,\rightarrow } and prefixing of quantifiers ∀v{\displaystyle \forall v} or ∃v{\displaystyle \exists v}. A sentence is a formula in which each occurrence of a variable is in the scope of a corresponding quantifier. Examples for formulas are φ (or φ(x) to mark the fact that at most x is an unbound variable in φ) and ψ defined as follows:

(Note that the equality symbol has a double meaning here.) It is intuitively clear how to translate such formulas into mathematical meaning. In the σsmr-structure N{\displaystyle {\mathcal {N}}} of the natural numbers, for example, an element nsatisfies the formula φ if and only if n is a prime number. The formula ψ similarly defines irreducibility. Tarski gave a rigorous definition, sometimes called "Tarski's definition of truth", for the satisfaction relation ⊨{\displaystyle \models }, so that one easily proves:

A set T of sentences is called a (first-order) theory. A theory is satisfiable if it has a modelM⊨T{\displaystyle {\mathcal {M}}\models T}, i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set T. Consistency of a theory is usually defined in a syntactical way, but in first-order logic by the completeness theorem there is no need to distinguish between satisfiability and consistency. Therefore, model theorists often use "consistent" as a synonym for "satisfiable".

A theory is called categorical if it determines a structure up to isomorphism, but it turns out that this definition is not useful, due to serious restrictions in the expressivity of first-order logic. The Löwenheim–Skolem theorem implies that for every theory T[2] which has an infinite model then for every infinite cardinal number κ, there is a model M⊨T{\displaystyle {\mathcal {M}}\models T} such that the number of elements of M{\displaystyle {\mathcal {M}}} is exactly κ. Therefore, only finitary structures can be described by a categorical theory.

Lack of expressivity (when compared to higher logics such as second-order logic) has its advantages, though. For model theorists, the Löwenheim–Skolem theorem is an important practical tool rather than the source of Skolem's paradox. In a certain sense made precise by Lindström's theorem, first-order logic is the most expressive logic for which both the Löwenheim–Skolem theorem and the compactness theorem hold.

As a corollary (i.e., its contrapositive), the compactness theorem says that every unsatisfiable first-order theory has a finite unsatisfiable subset. This theorem is of central importance in infinite model theory, where the words "by compactness" are commonplace. One way to prove it is by means of ultraproducts. An alternative proof uses the completeness theorem, which is otherwise reduced to a marginal role in most of modern model theory.

Axiomatizability, elimination of quantifiers, and model-completeness[edit]

The first step, often trivial, for applying the methods of model theory to a class of mathematical objects such as groups, or trees in the sense of graph theory, is to choose a signature σ and represent the objects as σ-structures. The next step is to show that the class is an elementary class, i.e. axiomatizable in first-order logic (i.e. there is a theory T such that a σ-structure is in the class if and only if it satisfies T). E.g. this step fails for the trees, since connectedness cannot be expressed in first-order logic. Axiomatizability ensures that model theory can speak about the right objects. Quantifier elimination can be seen as a condition which ensures that model theory does not say too much about the objects.

A theory T has quantifier elimination if every first-order formula φ(x1,...,xn) over its signature is equivalent modulo T to a first-order formula ψ(x1,...,xn) without quantifiers, i.e. ∀x1…∀xn(ϕ(x1,…,xn)↔ψ(x1,…,xn)){\displaystyle \forall x_{1}\dots \forall x_{n}(\phi (x_{1},\dots ,x_{n})\leftrightarrow \psi (x_{1},\dots ,x_{n}))} holds in all models of T. For example, the theory of algebraically closed fields in the signature σring=(×,+,−,0,1) has quantifier elimination because every formula is equivalent to a Boolean combination of equations between polynomials.

A substructure of a σ-structure is a subset of its domain, closed under all functions in its signature σ, which is regarded as a σ-structure by restricting all functions and relations in σ to the subset. An embedding of a σ-structure A{\displaystyle {\mathcal {A}}} into another σ-structure B{\displaystyle {\mathcal {B}}} is a map f: A → B between the domains which can be written as an isomorphism of A{\displaystyle {\mathcal {A}}} with a substructure of B{\displaystyle {\mathcal {B}}}. Every embedding is an injective homomorphism, but the converse holds only if the signature contains no relation symbols.

If a theory does not have quantifier elimination, one can add additional symbols to its signature so that it does. Early model theory spent much effort on proving axiomatizability and quantifier elimination results for specific theories, especially in algebra. But often instead of quantifier elimination a weaker property suffices:

A theory T is called model-complete if every substructure of a model of T which is itself a model of T is an elementary substructure. There is a useful criterion for testing whether a substructure is an elementary substructure, called the Tarski–Vaught test. It follows from this criterion that a theory T is model-complete if and only if every first-order formula φ(x1,...,xn) over its signature is equivalent modulo T to an existential first-order formula, i.e. a formula of the following form:

where ψ is quantifier free. A theory that is not model-complete may or may not have a model completion, which is a related model-complete theory that is not, in general, an extension of the original theory. A more general notion is that of model companions.

As observed in the section on first-order logic, first-order theories cannot be categorical, i.e. they cannot describe a unique model up to isomorphism, unless that model is finite. But two famous model-theoretic theorems deal with the weaker notion of κ-categoricity for a cardinal κ. A theory T is called κ-categorical if any two models of T that are of cardinality κ are isomorphic. It turns out that the question of κ-categoricity depends critically on whether κ is bigger than the cardinality of the language (i.e. ℵ0{\displaystyle \aleph _{0}} + |σ|, where |σ| is the cardinality of the signature). For finite or countable signatures this means that there is a fundamental difference between ℵ0{\displaystyle \aleph _{0}}-cardinality and κ-cardinality for uncountable κ.

Further, ℵ0{\displaystyle \aleph _{0}}-categorical theories and their countable models have strong ties with oligomorphic groups. They are often constructed as Fraïssé limits.

Michael Morley's highly non-trivial result that (for countable languages) there is only one notion of uncountable categoricity was the starting point for modern model theory, and in particular classification theory and stability theory:

If a first-order theory T in a finite or countable signature is κ-categorical for some uncountable cardinal κ, then T is κ-categorical for all uncountable cardinals κ.

Uncountably categorical (i.e. κ-categorical for all uncountable cardinals κ) theories are from many points of view the most well-behaved theories. A theory that is both ℵ0{\displaystyle \aleph _{0}}-categorical and uncountably categorical is called totally categorical.

Set theory (which is expressed in a countable language), if it is consistent, has a countable model; this is known as Skolem's paradox, since there are sentences in set theory which postulate the existence of uncountable sets and yet these sentences are true in our countable model. Particularly the proof of the independence of the continuum hypothesis requires considering sets in models which appear to be uncountable when viewed from within the model, but are countable to someone outside the model.

The model-theoretic viewpoint has been useful in set theory; for example in Kurt Gödel's work on the constructible universe, which, along with the method of forcing developed by Paul Cohen can be shown to prove the (again philosophically interesting) independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory.

In the other direction, model theory itself can be formalized within ZFC set theory. The development of the fundamentals of model theory (such as the compactness theorem) rely on the axiom of choice, or more exactly the Boolean prime ideal theorem. Other results in model theory depend on set-theoretic axioms beyond the standard ZFC framework. For example, if the Continuum Hypothesis holds then every countable model has an ultrapower which is saturated (in its own cardinality). Similarly, if the Generalized Continuum Hypothesis holds then every model has a saturated elementary extension. Neither of these results are provable in ZFC alone. Finally, some questions arising from model theory (such as compactness for infinitary logics) have been shown to be equivalent to large cardinal axioms.

A field or a vector space can be regarded as a (commutative) group by simply ignoring some of its structure. The corresponding notion in model theory is that of a reduct of a structure to a subset of the original signature. The opposite relation is called an expansion - e.g. the (additive) group of the rational numbers, regarded as a structure in the signature {+,0} can be expanded to a field with the signature {×,+,1,0} or to an ordered group with the signature {+,0,<}.

Similarly, if σ' is a signature that extends another signature σ, then a complete σ'-theory can be restricted to σ by intersecting the set of its sentences with the set of σ-formulas. Conversely, a complete σ-theory can be regarded as a σ'-theory, and one can extend it (in more than one way) to a complete σ'-theory. The terms reduct and expansion are sometimes applied to this relation as well.

Given a mathematical structure, there are very often associated structures which can be constructed as a quotient of part of the original structure via an equivalence relation. An important example is a quotient group of a group.

One might say that to understand the full structure one must understand these quotients. When the equivalence relation is definable, we can give the previous sentence a precise meaning. We say that these structures are interpretable.

A key fact is that one can translate sentences from the language of the interpreted structures to the language of the original structure. Thus one can show that if a structure M interprets another whose theory is undecidable, then M itself is undecidable.

Gödel's completeness theorem (not to be confused with his incompleteness theorems) says that a theory has a model if and only if it is consistent, i.e. no contradiction is proved by the theory. This is the heart of model theory as it lets us answer questions about theories by looking at models and vice versa. One should not confuse the completeness theorem with the notion of a complete theory. A complete theory is a theory that contains every sentence or its negation. Importantly, one can find a complete consistent theory extending any consistent theory. However, as shown by Gödel's incompleteness theorems only in relatively simple cases will it be possible to have a complete consistent theory that is also recursive, i.e. that can be described by a recursively enumerable set of axioms. In particular, the theory of natural numbers has no recursive complete and consistent theory. Non-recursive theories are of little practical use, since it is undecidable if a proposed axiom is indeed an axiom, making proof-checking a supertask.

The compactness theorem states that a set of sentences S is satisfiable if every finite subset of S is satisfiable. In the context of proof theory the analogous statement is trivial, since every proof can have only a finite number of antecedents used in the proof. In the context of model theory, however, this proof is somewhat more difficult. There are two well known proofs, one by Gödel (which goes via proofs) and one by Malcev (which is more direct and allows us to restrict the cardinality of the resulting model).

Model theory is usually concerned with first-order logic, and many important results (such as the completeness and compactness theorems) fail in second-order logic or other alternatives. In first-order logic all infinite cardinals look the same to a language which is countable. This is expressed in the Löwenheim–Skolem theorems, which state that any countable theory with an infinite model A{\displaystyle {\mathfrak {A}}} has models of all infinite cardinalities (at least that of the language) which agree with A{\displaystyle {\mathfrak {A}}} on all sentences, i.e. they are 'elementarily equivalent'.

Fix an L{\displaystyle L}-structure M{\displaystyle M}, and a natural number n{\displaystyle n}. The set of definable subsets of Mn{\displaystyle M^{n}} over some parameters A{\displaystyle A} is a Boolean algebra. By Stone's representation theorem for Boolean algebras there is a natural dual notion to this. One can consider this to be the topological space consisting of maximal consistent sets of formulae over A{\displaystyle A}. We call this the space of (complete) n{\displaystyle n}-types over A{\displaystyle A}, and write Sn(A){\displaystyle S_{n}(A)}.

Now consider an element m∈Mn{\displaystyle m\in M^{n}}. Then the set of all formulae ϕ{\displaystyle \phi } with parameters in A{\displaystyle A} in free variables x1,…,xn{\displaystyle x_{1},\ldots ,x_{n}} so that M⊨ϕ(m){\displaystyle M\models \phi (m)} is consistent and maximal such. It is called the type of m{\displaystyle m} over A{\displaystyle A}.

One can show that for any n{\displaystyle n}-type p{\displaystyle p}, there exists some elementary extensionN{\displaystyle N} of M{\displaystyle M} and some a∈Nn{\displaystyle a\in N^{n}} so that p{\displaystyle p} is the type of a{\displaystyle a} over A{\displaystyle A}.

Many important properties in model theory can be expressed with types. Further many proofs go via constructing models with elements that contain elements with certain types and then using these elements.

Illustrative Example: Suppose M{\displaystyle M} is an algebraically closed field. The theory has quantifier elimination . This allows us to show that a type is determined exactly by the polynomial equations it contains. Thus the space of n{\displaystyle n}-types over a subfield A{\displaystyle A} is bijective with the set of prime ideals of the polynomial ringA[x1,…,xn]{\displaystyle A[x_{1},\ldots ,x_{n}]}. This is the same set as the spectrum of A[x1,…,xn]{\displaystyle A[x_{1},\ldots ,x_{n}]}. Note however that the topology considered on the type space is the constructible topology: a set of types is basic open iff it is of the form {p:f(x)=0∈p}{\displaystyle \{p:f(x)=0\in p\}} or of the form {p:f(x)≠0∈p}{\displaystyle \{p:f(x)\neq 0\in p\}}. This is finer than the Zariski topology.

^In a countable signature. The theorem has a straightforward generalization to uncountable signatures.

^"All three commentators [i.e. Vaught, van Heijenoort and Dreben] agree that both the completeness and compactness theorems were implicit in Skolem 1923…." [Dawson, J. W. (1993). "The compactness of first-order logic:from gödel to lindström". History and Philosophy of Logic. 14: 15. doi:10.1080/01445349308837208.]